Yes, they are the same. I use "have to" as a casual (not important) saying. Example: You are speaking to your friends and you say "I have to get to the store today". If it is important you get to the store now, I would say "I must get to the store now". Hope that helps.

I suppose it's like German "sollen" (the analogue for "have to") and "mussen" (the analogue for "must").

Not exactly the same. When expressing a strong obligation, "must" suggests it is an obligation caused by the speaker's personal circumstances (and the speaker generally agrees with the obligation); while "have to" suggests some external circumstance makes the obligation necessary (which the speaker may be reluctant to perform). e.g. I don't have to finish the paper today because it is not due until next Friday, but I must as according to my own schedule.

No..In fact in school we've learned that it's wrong if u don't use "have to" "must" "should" or "might" in the right situation. "Must" is used for a law or when u wanna oblige someone to do or say something.Or when u r obliged,because of the facts of circumstances to do or say something.. "You must not(mustn't) smoke in the hospital" or "You must tell me" if that's very important for u and u wanna be more dramatic :D "have to" is when u just need to do something "You have to clean you're room" "Should" is when it would be better or more correct to do or say something or you're giving somebody an advice "You should tell her if there's something wrong wit you" "might" is when there are almost no possibility for this to happen or when u can't be sure about this. "It might rain today" ;)